Friday, 24 April 2015

Model of the Economy using Stocks and Flows. Part 2; Flows

A bit of background

The idea behind this model came from a result of some empirical research. I was using the dataset from my study on the effect of increased levels of private sector debt (if you haven't read it, the conclusion is that it is a bad thing), and I was empirically testing Michael Pettis's idea that a current account deficit must lead to a choice between unemployment and debt. 

I find Pettis' work to be truly exceptional and could happily read him all day. But the surprise to me came that I couldn't find as strong a link as I was expecting. At most, and this was setting all surpluses to a maximum of zero and isolating as best I could the effect of other variables, I found an approximate 0.5% increase in debt for every 1% of current account deficit. This was dwarfed by the average of 10% per year of new debt taken out in every country over the period.

Something else had to be going on, so I decided to model it.

Once again, I make the caveat that I am sure that this model can not be original. It is very straightforward. I just have not seen things put like this before, and I found it very helpful in understanding the flows in the economy. It is currently very simple but it could be subdivided further to make it more complex.

In terms of understanding the 'Flows' Part of the model, it would help to first be familiar with the 'Stocks' part that was the subject of my earlier post.

Update: I have just realised that I missed out the concept of Transfer of Existing Savings (TES) which I am now including below.


I will start with a closed system - a country with no trade or financial flows abroad. For simplicity, I will make it a discrete model - with action taking place once every year. This can be made to calculate every day, every minute, or (if someone were clever enough to make a differential equation) continuously.

At the beginning of every year all the money is paid out and then during the year it is either spent or saved. At the beginning of next year we start again.

The sources of money are: 

1) 'GDP' from work done the previous year.

2) 'New Cash' printed by the Central Bank, C. This could, in theory, be negative.

3) 'New Loans' created by private banks, L. This could also be written as New Deposits. This can be positive or negative.

4) Money taken 'From Existing Savings' to be spent in the economy. Eg pensioners spending their private pension. In terms of the Stocks model, the existing savings, ES,  can be seen as the sum of CB cash, equity and debt.

The GDP is divided into three flows:

a) Dividends - this includes eg. rental payments on houses. It is any return on equity as defined in my post on Stocks. This is denoted by d.

b) Interest, i.

c) Wages, w.

There is not much more required for this simple system in order to calculate the path of nominal GDP and total debt. Real GDP depends on actual productivity increases, but nominal GDP in this closed system is just how much money is spent (if there is more money spent but no increase in productivity then it is just inflation). To calculate the nominal GDP and net new savings/debt, I just need to put in how much is spent on consumption.

I will call α the coefficient of consumption. So αd is the proportion of the dividends paid that are spent in the economy. Similarly, αi, αwαL  and αC  are the coefficients of consumption of interest, wages, new loans and new cash. Also, I will define γ as the proportion of existing savings that are spent in the year.

We can then find the next year's nominal GDP and the amount of net savings by using the following:

GDP(t+1) = αd*d + αi*i + αw*w + αL*L + αC*C + γ*ES

And we can define Unspent Income (UI) as:

UI  = (1-αd)*d + (1-αi)*i + (1-αw)*w + (1-αL)*L + (1-αC)*C - γ*ES

Now, imagine that one year I decide not to polish the leather on my yacht as usual and save the money (£25k) instead. This £25k is a loss to GDP as I am not buying the services of the polisher. There are a number of ways that this can be made up. The polisher's employer could take out a loan and continue to pay the polisher. In this case the UI becomes a net new saving in the economy. The polisher's employer could continue to pay the polisher out of her savings, in which case there is a transfer of savings from the polisher's employer to myself. Or the polisher's employer could lay off the polisher in which case there would be some transfer from the polisher's savings to me and some taking out of new debt by the government to pay unemployment benefit.

So this UI can be split up into New Savings (NS) and Transfer of Existing Savings (TES).

We can now calculate NS as:

NS  = (1-αd)*d + (1-αi)*i + (1-αw)*w + (1-αL)*L + (1-αC)*C - γ*ES -TES

This can then all be plugged back into the equation for the next year. 

The reason, discussed here, that we are have a stagnating economy is that αw is α lot larger than αd and αi  and that w is too low as a proportion of GDP (equivalently d and i are too high). But one can show that there are other ways to boost GDP - for example by increasing to replace the demand lost by UI > 0.

Note that in the absence of new central bank cash, new net savings must always equal new debt. So for the next year, we have the following inputs, corresponding to the four listed above:

1) GDP => GDP(t+1) 

2) New Cash => C(t+1) 

3) New loans => Lt+1  NS - C(t+1) 

4) Existing Savings => ESt+1  = ESt  + NS

And the whole loop can start all over again (this model of existing savings is a little simple though - some sort of equity and bond price model allowing for economic growth should be included).

The most important part here is that for the system to be in what I call equilibrium, New Loans should be zero and reasonable growth should be achieved.

How does this fit in with the Stocks model?

In the stocks model, there are three quantities. These are the following:

1) Central Bank Cash, CBC.

2) Equity, E.

3) Debt, D.

I will also need to define the growth rate (which of course can be negative) of the value of existing debt and equity - these will be gdebt and geq.

The stocks are affected by the flows in the following ways:

1) CBC(t+1)  = CBCt + C(t+1) 

2)  E(t+1)  Et   (1+geq

3) D(t+1)  = Dt   (1+gdebt+  NS  -  C(t+1) 

Any other flows are just shifting cash around from one place to another. For example, if people generally thought that debt was too expensive compared to equity then gdebt might be negative and geq positive, but all that it would mean is money transferred around to different bank accounts. 

If money is used to improve the productive capacity of the economy then still it is just money moving around. However, one would expect to see this improvement reflected in the value of geq.

Adding the government to this

Note that I have ignored the government in this model, as all they do is spend a part of the wages, interest and dividends, and borrow a part of the New Loans. To put the government in to the model, one would need to first divert some of the wages, interest and dividends to government. It is also necessary to separate the New Loans into Government and Private Sector. Then calculate how much of that spending goes on wages, interest and dividends. 

With external trading partners

This was a closed system. Now let's put in an external trading partner - the world. For simplicity I will net everything off so that there is only an input of either positive or negative savings. This can be changed but just confuses the issue a bit here.

Let Foreign Savings (+ve or -ve) in year t be denoted by FS. This is also known as the current account balance. A current account deficit gives a positive value for FS, a surplus gives a negative value.

This is how the stocks are affected:

1) CBC(t+1)  = CBCt + C(t+1)

2)  E(t+1)  E (1+geq

3) D(t+1)  = Dt   (1+gdebt+ NS + FS - C(t+1) 

For this reason I argue in this post that we need to increase C to equal FS to counteract the effect of a positive FS.

It could also be argued that Ct+1 should be made to equal FS + NS so that debt no longer increases. In other words, new cash should be produced to replace the amount saved (or taken away to replace net spending of savings).

Anyway, regarding the foreign savings, this seems pretty straightforward, and it looks at first glance that it is an accounting identity that an increase in FS must by definition lead to a 1 to 1 increase in debt or a decrease in GDP as the domestic economy would have to suffer enough so that domestic savings go down to compensate. 

However, empirically I found this not to be the case. Why is this?

It is important to note where the current account deficit comes from. For example, imagine a situation where the only foreign holding in the domestic economy is of government bonds. And the deficit comes only from interest paid on holdings in government bonds. Then, looking at the flows above,  GDP will only fall by αi*deficit. Here, some of the domestic savings are being diverted abroad. New domestic saving will go down by (1 - αi)*deficit . And FS is 1* deficit - so it all balances. 

Alternatively, if the current account deficit is all due to traded goods then one might assume that this is taking 100% demand out of the economy. Thus the α can be considered to be 1. The rest of the UI is unaffected so the FS leads to increase in debt in a 1:1 ratio with the deficit.

So the composition of the current account deficit is important when trying to find out whether it will affect GDP or mainly just levels of domestic savings. 

Defining αFS as the coefficient of new foreign savings that would be spent in the economy gives us:

GDP(t+1) = αd*d + αi*i + αw*w + αL*L + αC*C + γ*ES - αFS*FS 

And we can define unspent income as:

UI = (1-αd)*d + (1-αi)*i + (1-αw)*w + (1-αL)*L + (1-αC)*C - γ*ES - (1-αFS)*FS 

And net new savings is still:



There isn't really a conclusion here; it is more of a framework that can be used to analyse different scenarios. However, if I were to draw one it would be that new central bank cash should be printed in enough quantity to offset the net domestic savings plus net foreign savings in UI.

This ensures that debt does not grow.  Depending on the α here, it should cause some inflation. Slowly this inflation will erode the value of the current debt until finally an equilibrium is reached. I will be modelling this in the near future so I will be able to report back more on the expected effects.


  1. Aside from the economic reasons for printing money - do you think it encourages 'too big to fail' banks to take even more risk?

  2. Stevie, there may be unforseen consequences - any policy like this should be started slowly.

    But I don't think that this increase in bank risk is one of them. The reason is that the attempt here is to shrink the size of the total debt. A rise in spending would probably need to be followed by rising interest rates to reduce the demand in the economy. As Central bank money goes up, private bank money will correspondingly reduce.

    If the total debt level is shrinking then bank risk will also go down.

  3. I agree, especially with regard to private debt, which has reduced drastically since QE. However, isn't the opposite true for banks' risk and liabilities? Haven't these increased in the last few years?

  4. The QE that we have seen so far is not what I am looking for. It is only the temporary replacing of government debt with cash.

    The inflation was caused at the time the government took out the debt, not when the proxy for cash (government bonds) was replaced by actual cash. In that sense, QE is just a huge cosmetic exercise.

    For QE to be effective at stimulating growth and inflation, the government debt would have to be permanently written off and the government would use that debt relief to increase spending. Then we would see real inflation and increase in economic growth.

    All we have seen with current QE, which is the only impact it can practically have, is a rise in asset prices. This then feeds through into a small increase in spending as people feel richer.

    Yes, this encourages more risk. And yes, this won't have made the banks more stable. But a real reduction in debt would.

  5. I think you're on the right track in your analysis of debt and its contribution to our economic woes. I have a few links to share , which you may already be aware of , but , in any case , are right down your alley. First , the BIS has a better data set on private debt than the OECD , IMO :

    Second , regarding second derivatives of debt , Biggs , Mayer , et al got that ball rolling. Here's one paper - you might be interested in the analysis starting on page 6 , where they try to determine the excess growth in debt/gdp for a given level of potential growth , and its implications for monetary policy :

    I share your frustration that mainstream economists haven't been giving this topic its due , but I see hopeful signs occasionally , like this recently from the St Louis Fed :


    1. Thank you Marko, this is very useful.

      Steve Keen actually very kindly also sent me that data, but the problem I have at the moment is that I also need corresponding levels of government debt in local currency, GDP in local currency and real GDP growth (not nominal) for that country for each year. And I am not sure where to find this.

      It is a project that I plan on completing, but at the moment, with the day job, I am finding it hard to get the time.

      I really appreciate the pointers to current research, and if you have any more suggestions I would be very interested.

    2. For data , I assume you're familiar with FRED. They have all of the categories you're interested in , though it takes a little work to find it sometimes. They might not be as complete in time series data as needed to fulfill your needs , but they add new data series all the time. I like it because you can easily graph and transform data ( within limits ) immediately.

      International data :

      BIS private debt series :

      Here's another recent paper that has similarities to your work :

      Look at fig 7 in particular. They derive "steady-state" measures of debt service ratios and net worth , and the plots show how deviations from these steady states impact real spending growth and real debt growth , for both the household (HS) and nonfinancial corporate (CS) sectors. They found that the increase in HS debt service in the 2000s offset the benefits of net worth increases , resulting in weak spending per dollar of new debt , as you've found.


    3. Marko, whoever you are, thank you. This is really a great help and that paper is excellent. Much appreciated and have a good day.



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